The concept of time scales analysis is a fairly new idea. In 1988, it was introduced by the German mathematician Stefan Hilger in his Ph.D. thesis and supervised by Bernd Aulbach. It combines the traditional areas of continuous and discrete analysis into one theory. After the publication of two textbooks in this area (by Bohner and Peterson, 2001, 2003) more and more researchers are getting involved in this fast-growing field of mathematics. Our paper summarizes the essentials of time scales calculus and hence helps researchers to become familiar with the subject and use these results for their own research.

  Does it describe a new discovery, methodology, or synthesis of knowledge?

Click for larger view “This calculus has potential applications in such areas as engineering, biology, economics and finance, physics, chemistry, social sciences, medical sciences, mathematics education, and others.”

The study of dynamic equations brings together the traditional research areas of (ordinary and partial) differential and difference equations. It allows one to handle these two research areas at the same time, hence shedding light on the reasons for their seeming discrepancies. In fact, many new results for the continuous and discrete cases have been obtained by studying the more general time scales case.

  Would you summarize the significance of your paper in layman’s terms?

In his book Men of Mathematics (Simon and Schuster, New York, pp. 13/14, 1937), Eric Temple Bell wrote: "A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both." Time scales analysis accomplishes exactly this. But not only can this theory unify continuous and discrete analysis, it is also able to extend the study of differential (continuous) and difference (discrete) equations to a more general class of dynamic equations, which includes, e.g., quantum-difference equations.

  The summary of our paper is as follows:

Section 1 gives an introduction, explains the unification and extension purpose of time scales analysis, and gives some important examples of time scales.

Section 2 introduces the delta derivative of a function defined on a time scale (this delta derivative is equal to the usual derivative for the continuous case, and it is equal to the usual forward difference for the discrete case).

Section 3 presents some special functions like the exponential function and trigonometric and hyperbolic functions on time scales and shows how they can be used to solve so-called dynamic equations on time scales.

Section 4 offers some examples and applications of the theory, including insect population models, which are discrete in season (and may follow a difference scheme with variable step-size or are often modeled by continuous dynamic systems), dying out in winter, while their eggs are incubating or dormant, and then, in season again, when hatching gives rise to a nonoverlapping population.

Sections 5 and 6 are devoted to linear dynamic equations and symplectic systems.

  How did you become involved in this research and were any particular problems encountered along the way?

All four of the authors of our paper are experts in both research areas of differential equations and of difference equations (all four have authored textbooks in the areas of both differential and difference equations; altogether, they have a current record of 1,269 publications). They therefore always were curious about unifying discrete and continuous analysis and were some of the first ones to realize that time scales analysis is the right tool for this goal. All four authors of our paper have been instrumental in presenting this subject to a wider audience of researchers on both national and international levels. On August 1, 2007, Professor Allan Peterson received the "2007 Euler Prize for Time Scales Research."

  Where do you see your research leading in the future?

Research in the area of dynamic equations on time scales is just at the beginning. Currently there are about 250 researchers worldwide who have published about 400 research articles in the area of time scales. The number of these researchers is steadily growing and this research area is wide open. This calculus has potential applications in such areas as engineering, biology, economics and finance, physics, chemistry, social sciences, medical sciences, mathematics education, and others.